two bodies of 9kg and 5.7kg are attached with ends of a string which passes over a pulley so that both bodies move vertically starting from rest at equal heights,lighter body hits pulley in 1.2sec.find length of string and speed with which it hits pulley.diameter of pulley is 14cm
ans (3.608m,3.388m\sec)
a stone is dropped from the top of a tower .in the last second of its motion it travels a distance of 44,1m.find(a)height of tower .(b)velocity while hitting the ground.
Let H be the tower heiight and g the acceleration of gravity (9.8 m/s^2). Let t be the time it takes to fall the full tower height. Here is what you know:
H = (g/2) t^2
H-44.1 = (g/2)(t-1)^2
Solve those two equations TOGETHER for the height H. It is easiest to solve for t first
44.1 = (g/2)*[t^2 - t^2 + 2t -1)
= 4.9 [2t - 1)
2t -1 = 9
t = 5 seconds
H = 4.9*25 = 122.5 m
(b) Vfinal^2 = 2 g H
To find the length of the string and the speed with which the lighter body hits the pulley, we will use some principles of physics.
Let's start by analyzing the motion of the lighter body:
1. Calculate the acceleration of the lighter body:
Since the lighter body hits the pulley in 1.2 seconds, we can find its acceleration using the equation:
Acceleration = Change in velocity / Change in time
In this case, the change in velocity is the final velocity of the lighter body, and the initial velocity is zero (as it started from rest). Therefore:
Acceleration = Final velocity / Change in time
We can assume the final velocity is v.
2. Calculate the net force acting on the lighter body:
The net force can be calculated using Newton's second law of motion:
Net Force = mass * acceleration
The mass of the lighter body is 5.7 kg.
3. Calculate tension in the string:
The tension in the string is the force exerted by the heavier body on the string. This tension is also equal to the weight of the lighter body. Therefore:
Tension = mass * acceleration due to gravity (9.8 m/s^2)
Now let's move on to analyzing the motion of the heavier body:
4. Calculate the acceleration of the heavier body:
Since both bodies are connected by a string, their accelerations are the same. Hence, the acceleration of the heavier body is also equal to the acceleration calculated in step 1.
Next, let's determine the length of the string:
5. Calculate the circumference of the pulley:
The circumference of the pulley can be calculated using the formula:
Circumference = π * diameter
The diameter of the pulley is given as 14 cm, so first we need to convert it to meters (since other measurements are given in meters).
6. Determine the length of the string:
As the string passes over the pulley, the length of the string is equal to the distance traveled by the lighter body.
Finally, let's calculate the speed at which the lighter body hits the pulley:
7. Calculate the final velocity of the lighter body:
The final velocity can be calculated using the formula:
Final velocity = Initial velocity + (Acceleration * Time)
Now, let's apply these steps to find the answers:
Step 1: Calculate the acceleration of the lighter body.
Acceleration = Final velocity / Change in time
Acceleration = v / 1.2s
Step 2: Calculate the net force acting on the lighter body.
Net Force = mass * acceleration
Net Force = 5.7kg * (v / 1.2s)
Step 3: Calculate tension in the string.
Tension = mass * acceleration due to gravity
Tension = 5.7kg * 9.8 m/s^2
Step 4: Calculate the acceleration of the heavier body.
Acceleration of heavier body = Acceleration of lighter body = v / 1.2s
Step 5: Calculate the circumference of the pulley.
Circumference = π * diameter
Circumference = π * 0.14m (converting 14cm to meters)
Step 6: Determine the length of the string.
The length of the string is equal to the distance traveled by the lighter body. As it falls, the distance traveled can be calculated using the formula:
Distance = 0.5 * Acceleration * Time^2
Distance = 0.5 * (v / 1.2s) * (1.2s)^2
Step 7: Calculate the final velocity of the lighter body.
Final velocity = Initial velocity + (Acceleration * Time)
Final velocity = 0 + (v / 1.2s) * 1.2s
The length of the string is obtained from step 6, and the speed with which the lighter body hits the pulley is obtained from step 7.